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G = C42.281C23  order 128 = 27

142nd non-split extension by C42 of C23 acting via C23/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42.281C23, C4:C4oSD16, C4oD8:7C4, D8:12(C2xC4), D8:C4:4C2, C4:C4.404D4, Q16:12(C2xC4), C4.154(C4xD4), Q16:C4:4C2, SD16:12(C2xC4), (C4xSD16):52C2, C8.24(C22xC4), C4.29(C23xC4), C22.22(C4xD4), C4:C4.369C23, C8o2M4(2):9C2, (C2xC8).420C23, (C4xC8).289C22, (C2xC4).209C24, C22:C4.191D4, C2.7(D4oSD16), D4.11(C22xC4), (C4xD4).60C22, C23.441(C2xD4), Q8.11(C22xC4), (C4xQ8).56C22, (C2xD4).377C23, (C2xD8).160C22, (C2xQ8).350C23, C8:C4.116C22, C4.Q8.129C22, C23.36D4:39C2, (C22xC4).930C23, (C22xC8).252C22, (C2xQ16).155C22, C22.153(C22xD4), D4:C4.199C22, C23.33C23:6C2, Q8:C4.200C22, (C2xSD16).178C22, C42:C2.300C22, (C2xM4(2)).356C22, C2.69(C2xC4xD4), (C2xC8):17(C2xC4), C4oD4:6(C2xC4), (C2xC4.Q8):8C2, C4.17(C2xC4oD4), (C2xC4oD8).16C2, (C2xC4).916(C2xD4), (C2xC4).268(C4oD4), (C2xC4:C4).577C22, (C2xC4).268(C22xC4), (C2xC4oD4).90C22, SmallGroup(128,1684)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C4 — C42.281C23
Chief series
C1 — C2 — C22 — C2xC4 — C22xC4 — C42:C2 — C23.33C23 — C42.281C23
Lower central
C1 — C2 — C4 — C42.281C23
Upper central
C1 — C22 — C42:C2 — C42.281C23
Jennings
C1 — C2 — C2 — C2xC4 — C42.281C23

Generators and relations for C42.281C23
 G = < a,b,c,d,e | a4=b4=1, c2=a2, d2=e2=b2, ab=ba, ac=ca, ad=da, eae-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=b2c, de=ed >

Subgroups: 404 in 242 conjugacy classes, 140 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, M4(2), D8, SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C4oD4, C4xC8, C8:C4, D4:C4, Q8:C4, C4.Q8, C2xC4:C4, C2xC4:C4, C42:C2, C42:C2, C4xD4, C4xD4, C4xQ8, C22xC8, C2xM4(2), C2xD8, C2xSD16, C2xQ16, C4oD8, C2xC4oD4, C8o2M4(2), C23.36D4, C2xC4.Q8, C4xSD16, Q16:C4, D8:C4, C23.33C23, C2xC4oD8, C42.281C23
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22xC4, C2xD4, C4oD4, C24, C4xD4, C23xC4, C22xD4, C2xC4oD4, C2xC4xD4, D4oSD16, C42.281C23

Smallest permutation representation of C42.281C23
►On 64 points
Generators in S64
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)

(1 15 27 23)(2 16 28 24)(3 13 25 21)(4 14 26 22)(5 11 35 31)(6 12 36 32)(7 9 33 29)(8 10 34 30)(17 42 49 62)(18 43 50 63)(19 44 51 64)(20 41 52 61)(37 53 57 45)(38 54 58 46)(39 55 59 47)(40 56 60 48)

(1 37 3 39)(2 38 4 40)(5 52 7 50)(6 49 8 51)(9 43 11 41)(10 44 12 42)(13 47 15 45)(14 48 16 46)(17 34 19 36)(18 35 20 33)(21 55 23 53)(22 56 24 54)(25 59 27 57)(26 60 28 58)(29 63 31 61)(30 64 32 62)

(1 11 27 31)(2 12 28 32)(3 9 25 29)(4 10 26 30)(5 23 35 15)(6 24 36 16)(7 21 33 13)(8 22 34 14)(17 58 49 38)(18 59 50 39)(19 60 51 40)(20 57 52 37)(41 53 61 45)(42 54 62 46)(43 55 63 47)(44 56 64 48)

(1 31 27 11)(2 12 28 32)(3 29 25 9)(4 10 26 30)(5 23 35 15)(6 16 36 24)(7 21 33 13)(8 14 34 22)(17 46 49 54)(18 55 50 47)(19 48 51 56)(20 53 52 45)(37 41 57 61)(38 62 58 42)(39 43 59 63)(40 64 60 44)
 
G:=sub<Sym(64)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,15,27,23)(2,16,28,24)(3,13,25,21)(4,14,26,22)(5,11,35,31)(6,12,36,32)(7,9,33,29)(8,10,34,30)(17,42,49,62)(18,43,50,63)(19,44,51,64)(20,41,52,61)(37,53,57,45)(38,54,58,46)(39,55,59,47)(40,56,60,48), (1,37,3,39)(2,38,4,40)(5,52,7,50)(6,49,8,51)(9,43,11,41)(10,44,12,42)(13,47,15,45)(14,48,16,46)(17,34,19,36)(18,35,20,33)(21,55,23,53)(22,56,24,54)(25,59,27,57)(26,60,28,58)(29,63,31,61)(30,64,32,62), (1,11,27,31)(2,12,28,32)(3,9,25,29)(4,10,26,30)(5,23,35,15)(6,24,36,16)(7,21,33,13)(8,22,34,14)(17,58,49,38)(18,59,50,39)(19,60,51,40)(20,57,52,37)(41,53,61,45)(42,54,62,46)(43,55,63,47)(44,56,64,48), (1,31,27,11)(2,12,28,32)(3,29,25,9)(4,10,26,30)(5,23,35,15)(6,16,36,24)(7,21,33,13)(8,14,34,22)(17,46,49,54)(18,55,50,47)(19,48,51,56)(20,53,52,45)(37,41,57,61)(38,62,58,42)(39,43,59,63)(40,64,60,44)>;
 
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,15,27,23)(2,16,28,24)(3,13,25,21)(4,14,26,22)(5,11,35,31)(6,12,36,32)(7,9,33,29)(8,10,34,30)(17,42,49,62)(18,43,50,63)(19,44,51,64)(20,41,52,61)(37,53,57,45)(38,54,58,46)(39,55,59,47)(40,56,60,48), (1,37,3,39)(2,38,4,40)(5,52,7,50)(6,49,8,51)(9,43,11,41)(10,44,12,42)(13,47,15,45)(14,48,16,46)(17,34,19,36)(18,35,20,33)(21,55,23,53)(22,56,24,54)(25,59,27,57)(26,60,28,58)(29,63,31,61)(30,64,32,62), (1,11,27,31)(2,12,28,32)(3,9,25,29)(4,10,26,30)(5,23,35,15)(6,24,36,16)(7,21,33,13)(8,22,34,14)(17,58,49,38)(18,59,50,39)(19,60,51,40)(20,57,52,37)(41,53,61,45)(42,54,62,46)(43,55,63,47)(44,56,64,48), (1,31,27,11)(2,12,28,32)(3,29,25,9)(4,10,26,30)(5,23,35,15)(6,16,36,24)(7,21,33,13)(8,14,34,22)(17,46,49,54)(18,55,50,47)(19,48,51,56)(20,53,52,45)(37,41,57,61)(38,62,58,42)(39,43,59,63)(40,64,60,44) );
 
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,15,27,23),(2,16,28,24),(3,13,25,21),(4,14,26,22),(5,11,35,31),(6,12,36,32),(7,9,33,29),(8,10,34,30),(17,42,49,62),(18,43,50,63),(19,44,51,64),(20,41,52,61),(37,53,57,45),(38,54,58,46),(39,55,59,47),(40,56,60,48)], [(1,37,3,39),(2,38,4,40),(5,52,7,50),(6,49,8,51),(9,43,11,41),(10,44,12,42),(13,47,15,45),(14,48,16,46),(17,34,19,36),(18,35,20,33),(21,55,23,53),(22,56,24,54),(25,59,27,57),(26,60,28,58),(29,63,31,61),(30,64,32,62)], [(1,11,27,31),(2,12,28,32),(3,9,25,29),(4,10,26,30),(5,23,35,15),(6,24,36,16),(7,21,33,13),(8,22,34,14),(17,58,49,38),(18,59,50,39),(19,60,51,40),(20,57,52,37),(41,53,61,45),(42,54,62,46),(43,55,63,47),(44,56,64,48)], [(1,31,27,11),(2,12,28,32),(3,29,25,9),(4,10,26,30),(5,23,35,15),(6,16,36,24),(7,21,33,13),(8,14,34,22),(17,46,49,54),(18,55,50,47),(19,48,51,56),(20,53,52,45),(37,41,57,61),(38,62,58,42),(39,43,59,63),(40,64,60,44)]])
 

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4L4M···4X8A8B8C8D8E···8J
order12222222224···44···488888···8
size11112244442···24···422224···4

44 irreducible representations

dim11111111112224
type+++++++++++
imageC1C2C2C2C2C2C2C2C2C4D4D4C4oD4D4oSD16
kernelC42.281C23C8o2M4(2)C23.36D4C2xC4.Q8C4xSD16Q16:C4D8:C4C23.33C23C2xC4oD8C4oD8C22:C4C4:C4C2xC4C2
# reps112142221162244

Matrix representation of C42.281C23 ►in GL6(F17)

1300000
0130000
004400
0091300
00001313
000084
,
1600000
0160000
00161600
002100
00001616
000021
,
620000
7110000
000063
00001111
006300
00111100
,
1600000
610000
00001616
000001
001100
0001600
,
1600000
0160000
000010
000001
0016000
0001600

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,9,0,0,0,0,4,13,0,0,0,0,0,0,13,8,0,0,0,0,13,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,2,0,0,0,0,16,1,0,0,0,0,0,0,16,2,0,0,0,0,16,1],[6,7,0,0,0,0,2,11,0,0,0,0,0,0,0,0,6,11,0,0,0,0,3,11,0,0,6,11,0,0,0,0,3,11,0,0],[16,6,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,16,0,0,16,0,0,0,0,0,16,1,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

C42.281C23 in GAP, Magma, Sage, TeX 

C_4^2._{281}C_2^3
 
% in TeX
 
G:=Group("C4^2.281C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1684);
 
// by ID
 
G=gap.SmallGroup(128,1684);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,184,2019,521,248,2804,1411,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=a^2,d^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b^2,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,e*c*e^-1=b^2*c,d*e=e*d>;
 
// generators/relations
 

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